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4. Groups of 1

4. Groups of permutations

Consider a of n distinguishable objects, {B1,B2,B3,...,Bn}. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given operation is 1 2 3 . . . n ! , a1 a2 a3 . . . an where the numbers {a1, a2, a3, . . . , an} are the numbers {1, 2, 3, . . . , n} in some . This operation can be interpreted in two different ways, as follows.

Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. . . , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set.

Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. . . , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical – for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD.

Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant – the columns may be written in any order without affecting the result, provided each column is kept intact. The chronological order in which the objects are rearranged doesn’t matter. 1 2 3 4 ! As an example, consider the action of the permutation 4 1 3 2 on the ordered set wxyz. Using interpretation 1, the first object, w, is moved to the 4th position; the second object, x, to the 1st position; the third object, y, to the 3rd position; the fourth object, z, to the 2nd position, producing the final order xzyw. In order to use interpretation 2, the objects in the

Introductory for Physicists Michael W. Kirson 4. Groups of permutations 2

set must be relabeled — x1 = w, x2 = x, x3 = y, x4 = z — after which the label 1 is replaced by 4, the label 2 by 1, the label 3 by 3 and the label 4 by 2, producing the final order x4x1x3x2 = zwyx. Both interpretations lead to well-defined permutations of the symbols w, x, y, z, but give different results, so it is essential to specify the interpretation being used. Now consider the effect of multiplying permutations, where multiplication is defined as consecutive action of two permutations. (Recall the standard convention that the right-hand factor in a product acts first, followed by the left-hand factor.) On the result of the above example, act with the permu- 1 2 3 4 ! tation . Using interpretation 1, it acts on xzyw to produce 4 3 2 1 1 2 3 4 ! wyzx, which is produced from wxyz by the permutation . 1 4 2 3 1 2 3 4 ! 1 2 3 4 ! 1 2 3 4 ! This multiplication can be written = . 4 3 2 1 4 1 3 2 1 4 2 3 By reordering the columns in the left-hand factor of the product so that the top row reads the same as the bottom row of the right-hand factor, this can 4 1 3 2 ! 1 2 3 4 ! 1 2 3 4 ! be rewritten in the form = . 1 4 2 3 4 1 3 2 1 4 2 3 In this form, it is evident that multiplication can be carried out by “cancel- ing” the bottom row of the right-hand factor against the identical top row of the left-hand factor. The result has the top row of the right-hand factor and the bottom row of the left-hand factor. A moment’s consideration will show that this is a universal feature (it simply says, for example, move the 1st object to the 4th position in the first step and the 4th object to the 1st position in the second step, leaving the 1st object in the 1st position overall, and so on), which makes multiplication of permutations trivial. The same process can be carried out using interpretation 2. Now the second permutation acts on zwyx (which is x4x1x3x2) to produce x1x4x2x3, or wzxy, which is also the result of acting on wxyz with the permutation 1 2 3 4 ! . The multiplication is expressed in permutation symbols in 1 4 2 3 exactly the same way as in interpretation 1, and the same rule of multiplica- tion by cancellation can be applied. [The two interpretations again lead to different final results, wyzx and wzxy, even though they are applied to the same symbolic multiplication.] The cancellation rule for evaluating products of permutations shows triv- ially that the multiplication is associative. There is an identity permutation

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 3

1 2 . . . n ! , and the inverse of any permutation is obtained by ex- 1 2 . . . n changing its top and bottom rows. So these transformations constitute a , where multiplication is understood as consecutive action of the trans- formations.

1. A permutation of the set {Bi}, followed by another permutation, clearly produces a permutation of the original set, so the permutations are closed under multiplication. 2. The multiplication of permutations has been shown to be associative. (A permutation can be regarded as a mapping of the set of ordered n-tuples of , confirming that the multiplication is associative.) 3. The permutation which leaves the ordered set unchanged is clearly an identity.

4. Given a permutation which changes an initial ordering of the set {Bi} into a final ordering, the permutation which changes the final ordering back into the initial ordering is the inverse of the starting permutation.

So the set of all permutations of n objects forms a group, called the on n objects, denoted Sn. It has order n!. An alternative, more economical and very efficient notation for permu- tations is provided by cycles.A cycle is a sequence of up to n labels, (a1, a2, a3, . . . , am)(m ≤ n) and represents a permutation symbol a a a . . . a a b . . . b ! 1 2 3 m−1 m 1 n−m , a2 a3 a4 . . . am a1 b1 . . . bn−m where {bi} are all the n − m labels not included in the set {ai}. It permutes the labels {ai} cyclically and leaves the labels {bi} unchanged. The number of labels it contains is the degree of the cycle. A cycle of degree m is referred to as an m-cycle. It is easily seen that two cycles with no label in common commute with one another, while two cycles with any labels in common will generally not commute with one another. The order of the entries in a cycle is significant, but since it is cyclic, the starting point is irrelevant: (a1, a2, . . . , am) = (a2, . . . , am, a1), etc., so an m-cycle can be written in m different but equivalent ways. For example, (a1, a2, a3, a4) = (a2, a3, a4, a1) = (a3, a4, a1, a2) = (a4, a1, a2, a3). The order of a cycle is equal to its degree — acting m times with an m- cycle restores all labels to their starting positions, i.e. is equal to the identity.

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 4

Any 1-cycle is equal to the identity, as is any product of 1-cycles. A 2-cycle is called a transposition, corresponds to interchanging its two entries, and is its own inverse. Any cycle can be decomposed into a (non-unique) product of transpositions. For instance, (1, 2, 3, . . . , n) = (1, n)(1, n − 1) ... (1, 3)(1, 2). [This is not unique because the product (a, b)(a, b) = 1 can be inserted any- where.] Any permutation can be uniquely resolved into a product of non-overlapping 1 2 3 . . . n ! cycles — in the permutation , rearrange the columns a1 a2 a3 . . . an 0 ! 1 a1 . . . am−1 ... 0 so that it starts = (1, a1, . . . , am−1); then rear- a1 aa1 ... 1 ... range the remaining columns in similar cyclic order, beginning with a label 0 not in the set {1, a1, aa1 , . . . , am−1}; and continue until all labels are ex- hausted. It is customary to omit 1-cycles from this resolution, which is self-evidently unique. The result of the resolution can be characterised by listing the number of r-cycles, νr, for each value of r from 1 to n. The list {νr} defines the cycle structure of the permutation. A permutation which is resolved into cycles all of the same degree is called a regular permutation and its order is equal to the degree of its cycles. A regular permutation has either only 1-cycles, in which case it is the identity permutation, or no 1-cycles, in which case it leaves no symbol unpermuted.

The symmetric group on n objects Sn, of order n!, has many , each of which is a group of permutations. A group of permutations is regular if all its elements are regular. If all the elements of a group of permutations, except the identity, leave no symbol unpermuted, then the group is regular. [Suppose a permutation contains two cycles of different degrees, k < l. Then k applications of the permutation leave k symbols unpermuted and l symbols permuted. But the group is closed under multiplication, so k applications of any must leave all or none of the symbols unpermuted.]

 {λ}  The conjugate of a permutation by some permutation T = {λ0} is the product of the conjugates of its constituent cycles. The conjugate of a cycle 0 0 0 0 0 C = (a1, a2, . . . , am) is the cycle C = (a1, a2, . . . , am), where ai is the symbol into which ai is changed by the permutation T .

−1 0 [Suppose a particular λ is not one of the {ai}. Then T : λ 7→ 0 −1 0 0 λ; C : λ 7→ λ; T : λ 7→ λ =⇒ TCT : λ 7→ λ . If λ = ai for −1 0 0 some i, then T : ai 7→ ai; C : ai 7→ ai+1; T : ai+1 7→ ai+1 =⇒ −1 0 0 TCT : ai 7→ ai+1.]

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 5

So conjugation turns an m-cycle into another m-cycle, which implies that all elements of a given class have the same cycle structure (i.e. the same number of cycles of the same degrees).

For the symmetric group on n objects, Sn, the converse also holds — all permutations having the same cycle structure belong to the same class.

0 0 0 [For two permutations C1C2 ...Ck and C1C2 ...Ck, where the cy- 0 cles Ci and C have the same degree for all i, conjugate with the i ! C1 C2 ...Ck permutation T = 0 0 0 , where each cycle in the C1 C2 ...Ck permutation symbol is replaced by the string of labels it contains. The permutation T is an element of Sn, so the original two per- mutations belong to the same class. The result does not hold for a general permutation group, since such a group will generally not contain all the permutations T required for conjugation.]

So a class of Sn is fully defined by its cycle structure — νr cycles of each Pn degree r from 1 to n, such that r=1 rνr = n. The number of elements in a class is the number of different ways of dividing n symbols into r-cycles, Qn νr where each r-cycle occurs νr times, with 1 ≤ r ≤ n, namely n!/ r=1 r νr!. [Let the cycles be placed side by side and filled with n symbols in all possible ways. There are n! such arrangements. All ar- rangements in which the νr distinct r-cycles are permuted among themselves are equivalent, so the total number must be divided by the product of νr!. All arrangements in which the entries in a particular r-cycle are cyclically permuted are equivalent, so the total number must be divided by the product of rνr .]

Consider the set of numbers

n X µi = νj. (1) j=i

These satisfy µ1 ≥ µ2 ≥ µ3 ≥ ... ≥ µn ≥ 0 (2)

µi − µi+1 = νi (µn = νn =⇒ µn+1 = 0) (3) n X µi = n (4) i=1 so they constitute a partition of n which specifies completely a given class of Sn. The classes of Sn are defined by the partitions of n, one to each

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 6 partition, and the number of elements in a class is given by the expression quoted above. Consider the formal n Y ∆ = (xi − xj), i

Since ζ(P1P2)∆ = P1P2∆ = ζ(P2)P1∆ = ζ(P2)ζ(P1)∆, it follows that −1 −1 ζ(P1P2) = ζ(P1)ζ(P2). Clearly ζ(1) = 1, so ζ(P ) = ζ(P ). Also ζ(P2P1P2 ) = ζ(P1), so all members of the same class have the same alternating character. Since ζ((1, 2)) = −1, by inspection, all transpositions are odd. From the de- composition of a cycle into a product of transpositions, ζ((a1, a2, . . . , am)) = m−1 (−) . For the class defined by the partition {µi}, with the associated {νi}, P ν (r−1) n−µ the alternating character is (−) r r = (−) 1 . In any group G of permutations (not necessarily the full symmetric group Sn), either half of the permutations are even or all of them are. P [If the group contains any odd permutation, Q say, then P ∈G ζ(P ) = P P P P ∈G ζ(PQ) = P ∈G ζ(P )ζ(Q) = − P ∈G ζ(P ) = 0, where the rearrangement theorem has been used in the first step.]

The set of even permutations of Sn is a , of order n!/2, called the An. The multiplication table of a finite group G has the property that ev- ery row is a permutation of the first row, with the additional limitation that no element of the group can occur in the same column in different rows. There are n such permutations for a group of order n. Each ele- ment of the group can be associated with the permutation defined by the row of the multiplication table which starts with that element. Let the el- ements of the group be {G1,G2,G3,...,Gn}, where usually G1 = E, the of G. The permutation associated with an element X ∈ G is G G G ...G ! 1 2 3 n , where interpretation 2 must necessarily XG1 XG2 XG3 ...XGn

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 7

be used – the permutation replaces each element Gi by the element XGi of the group, regardless of its position in the list of elements. (Recall that the rearrangement theorem ensures that the bottom row of this symbol is a permutation of the top row.) Consider the product of two such permutations, using the abbreviated {G } ! notation i for a permutation. Note that, by the rearrangement {XGi} {YG } ! {G } ! theorem, i = i , for any element Y ∈ G, since the {XYGi} {XGi} two permutation symbols differ only in the order of their columns. So {G } ! {G } ! {YG } ! {G } ! {G } ! i i = i i = i , by {XGi} {YGi} {XYGi} {YGi} {XYGi} the cancellation rule for multiplication of permutation symbols. By closure {G } ! for G, the product XY ∈ G, so the permutation i is one of the {XYGi} set of n permutations associated with elements of G. This set is therefore closed under multiplication, which is the usual multiplication of permuta- tions and hence associative. (This same result shows that the association of permutations with elements of the group G preserves the multiplication.) −1 The existence of an identity E and inverses {Gi } in G then ensures the ex- {G } ! {G } ! istence of an identity permutation i = i and inverse per- {EGi} {Gi} ! {Gi} mutations −1 , so the set of associated permutations is a group. It {X Gi} is of order n and every one of its permutations is associated with an element of the group G. The association defined above is therefore a mapping from G onto the group of associated permutations, which preserves multiplication, i.e. it is a {G } ! homomorphism. But by the properties of permutation symbols, i = {XGi} ! {Gi} only if XGi = YGi for all Gi ∈ G, i.e. only if X = Y (by the {YGi} cancellation property of G). So the mapping under discussion is 1–1 and the homomorphism is an . ! {Gi} The permutation replaces the element Gi ∈ G by XGi ∈ {XGi} 2 2 3 {Gi}, the element XGi by X Gi, the element X Gi by X Gi, and so on, r−1 r until X Gi is replaced by X Gi = Gi, where r is the order of X in G.

Introductory Algebra for Physicists Michael W. Kirson 4. Groups of permutations 8

2 r−1 Thus the permutation contains an r-cycle (Gi,XGi,X Gi,...,X Gi). The cycle could not be of degree less than r, since that would contradict the statement that X is of order r. Suppose, for some j, the element Gj does 2 r−1 not belong to this r-cycle. Then (Gj,XGj,X Gj,...,X Gj) is another r-cycle contained in the permutation associated with X, with no overlap with the first cycle. This argument may be repeated until all elements Gi are exhausted. So the permutation may be decomposed into r-cycles and is hence regular. The result holds for every X ∈ G, with its appropriate order, so the group of permutations isomorphic to G is regular. This is frequently referred to as the of G. (The fact that the group is regular can be more directly established by noting that the rearrangement theorem ensures that every permutation other than the identity leaves no elements of G unpermuted.) These results prove Cayley’s Theorem: every finite group of order n is isomorphic to a regular subgroup of the symmetric group Sn of order n!.

Introductory Algebra for Physicists Michael W. Kirson